Geodesic
ON THE NUMBER OFCRITICAL POINTS OF A C1 FUNCTION ON THE SPHERE
Glasgow mathematical journal, Tome 42 (2000) no. 2, pp. 283-287

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For a C1function f:R^n →R\;(n \ge 2), we consider the least numberk of distinct critical points that f must possess whenrestricted to the sphere S=\{x\in R^n: \Vert x\Vert =1\}. Clearly k\ge 2 (for f attains its absolute minimum and maximum onS), and a result of Lusternik and Schnirelmann establishes thatk=n if f is even. Here we prove that k=n if,for a given orthonormal system (e_i), \max\limits_{S \capV_i}\,f<\min\limits_{S \cap V_i^\bot}\,f, for all i=1, ...n-1,where V_i is the subspace spanned by e_1, ..., e_i andV_i^\bot its orthogonal complement. It is shown that this criterion is satisfied bysuitably restricted perturbations of quadratic forms having n distincteigenvalues. 
ON THE NUMBER OFCRITICAL POINTS OF A C1 FUNCTION ON THE SPHERE. Glasgow mathematical journal, Tome 42 (2000) no. 2, pp. 283-287. doi: 10.1017/S0017089500020152
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     journal = {Glasgow mathematical journal},
     pages = {283--287},
     year = {2000},
     volume = {42},
     number = {2},
     doi = {10.1017/S0017089500020152},
     url = {http://geodesic.mathdoc.fr/articles/10.1017/S0017089500020152/}
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